Well, the first thing you have done wrong is (x+2)^2 does not equal to (x+2)+(x+2), it's (x+2)*(x+2)
so x^2+(x+2)*(x+2)=290
x^2+x^2+4x+4=290
2x^2+4x+4=290
2x^2+4x-286=0
now, you can either solve this by the quadradic formula, or factoring
QF: x=[-4±√(16+2288)]/4
x=(-4±48)/4
x=11 or -13
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on your work
if you did x^2 + (x +2) + (x + 2) = 290, it should be x^2+2x+4, how did you get 2x^2 + 4x + 4 = 290?
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Check: plug in 11
121+169=290
169+121=290